Find the sum of $14 + 8 + 2 +... + (-274) + (-280)$.
Solution: Getting started We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $6$ less than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {14})$ and the last term $(a_n = {-280})$ are given in the question. We need to find $n$ (the number of terms). Step 1: Find $n$ (the number of terms) The sequence decreases by $14 - (-280) = 294$ from the first term to the last term. Because the sequence decreases by $6$ each time, it takes $\dfrac{294}{6} = 49$ terms to get from the first term to the last term. We still need to count the first term, so there are $49 + 1 = {50}$ terms in the sequence. In other words, $n = {50}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{50}}&= \dfrac {\left({14} + ({-280}) \right)}{2} \cdot {50} \\\\ S_{{50}} &= -133 \left(50\right) \\\\ S_{{50}} &= -6650\end{aligned}$ The answer $ -6650 $